Represented records for three named scalar-field models Finite-plane corrections, dense-TT corrections, and sourced curvature pairings

We construct represented binary measurements in three separate models. On a fixed nondegenerate transverse-traceless plane, a first-order inhomogeneous scalar correction becomes a quadratic oscillator–field operator. It is essentially self-adjoint, a smooth initial-time-edge regulator can be removed, and two gravitational coherent preparations with the same scalar factor yield a positive Ramsey contrast. On an infinite-dimensional vacuum transverse-traceless completion, the analogous bilinear functional has explicit four-leg kernels. Their spectral masks can be removed in operator norm after a fixed particle-number compression, with separate Hilbert–Schmidt and operator-norm coefficient estimates and a controlled record remainder. Independently, a specified Newton–wave response to normal-ordered massive scalar stress defines a metric-valued form on scalar Fock space. A compact exterior curvature pairing gives a positive vacuum–coherent record at fixed scalar momentum cutoff and sufficiently large finite number cutoff. Every record theorem includes an exact zero-interpolation control. The three representations, sources, and cutoffs are not identified. The measured dense-TT and sourced models retain their number cutoffs, and all apparatuses are global controlled-unitary instruments.

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Classical sourced $j$ exists. It is not an argument of $A^{(1)}$ or $A^{(2)}$, and it is not the cubic $S_{t_0}(j)\varphi$.

The question is whether a specified classical functional admits a complete chain

Each arrow requires its own definition and proof. A classical gauge-invariant expression does not automatically define an operator. A symmetric form cannot automatically be exponentiated. A unitary system channel does not by itself specify its controlled realization.

We answer this question in three named models. They are three models, not three approximations to one operator.

Figure fig:three-models separates the three classical quantities, representations, generators, and record endpoints.

The third quantity is not the first-order scalar correction of the first two models. It is a quadratic scalar composite obtained by reconstructing a sourced metric coefficient and smearing its linearized curvature. Its independent inclusion is the reason for the broader title.

The numerical parameter $\lambda \in \mathbb{R}$ belongs to each specified measurement generator. It interpolates between that generator and its declared control. It is not identified with Newton's constant, nor with evaluation of a nilpotent formal parameter at a real number.

The distinction between an identified measurement semantics and a represented implementation is also present in the Enter calculus EnterCalculus. We use no measurement-uniqueness theorem here. In particular, the infinite-dimensional operators below are not asserted to be primitives of a finite-qubit language.

The symbols have different meanings:

$\epsilon$, Model 1.

A smooth replacement of a sharp initial-time edge in a scalar smearing. It is removed at the unitary and record levels.

$R$, Model 2.

A simultaneous gravitational infrared and ultraviolet mask and a scalar ultraviolet mask. It is removed at fixed number cutoff $M$.

$M$, Models 2 and 3.

A particle-number projection on the Hilbert space of that model. It remains finite in every measured evolution considered here. Statements uniform for sufficiently large $M$ do not assert an instrument limit as $M\to \infty$.

$\Lambda$, Model 3.

A scalar momentum cutoff. It is retained in its measurement theorem. A classical or coherent expectation limit is used only to select one sufficiently large finite value.

$N$, Model 2.

A finite-rank projection on gravitational one-particle kernel legs. It is not a particle-number cutoff.

No theorem constructs a full reduced Einstein–scalar phase space, a sourced graviton representation, an unregulated cubic scalar correction, or a compact system–probe field coupling. Models 1 and 2 remain vacuum gravitational models. Model 3 has no independent quantum metric degree of freedom. The controlled-unitary apparatus is admitted globally in each model.

Figure fig:cutoff-grammar records which regulators are removed and which remain in the measurement theorems.

Theorem thm:model1 gives the Model 1 record, including removal of $\epsilon$ and exact equality at $\lambda =0$. Theorem thm:model2 gives the Model 2 fixed-$M$ spectral limit and uniformly calibrated contrast. Theorem thm:model3 gives the Model 3 record at fixed $\Lambda$ and finite $M$, with both interpolation-off and source-off controls.

The classical constructions needed to identify the functionals are included in their stated domains. No result about a larger equation space is imported.

We use $\hslash =1$, signature $(-1,1,1,1)$, and

A fixed reference time is denoted by $t_0$. The spatial Fourier transform is

Inner products are antilinear in their first entry. Tensor contraction is included in inner products of tensor-valued functions.

The scalar Fock convention is

so $\Phi$ is real linear and

The normalized coherent vector is

Its number distribution is Poisson with mean ${\Vert z\Vert }^2$.

For a smooth symmetric tensor $h$, let

and define

For $(K\xi {)}_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$ and $X_{\xi }={\xi }^{\mu }{\partial }_{\mu }$,

These identities follow by differentiating the displayed formulas; the second also follows by differentiating scalar covariance under a diffeomorphism.

For a free scalar $P\varphi =0$, set

It solves

For $b\in L^1(I;H^s({\mathbb{R}}^3))$, $s\ge 0$, and a compact time interval $I$ containing $t_0$, the zero-data solution of $({{\partial }_t}^2-\Delta +m^2)u=b$ is the sine integral in (eq:S) with $b$ in place of $L_h\varphi$, and

It is unique in this class. Smooth data and source give a smooth solution. Propagation has speed at most one.

Proof. The Fourier multipliers are $\sin ((t-s){\omega }_p)/{\omega }_p$ and $\cos ((t-s){\omega }_p)$, with ${\omega }_p=\sqrt{|p{|}^2+m^2}$. Their weighted bounds give the estimate and differentiation proves the equation. Uniqueness follows from the homogeneous energy identity. For finite propagation integrate

over a truncated causal cone. In a source-free cone the outward characteristic flux is nonnegative; zero initial energy implies zero solution there. Smoothness follows by applying the estimates at all orders.

If $\varphi$ has compact smooth data, its support is spatially bounded on every finite slab. Thus (eq:S) is well defined for every smooth $h$, even one with a Coulomb tail: $L_h$ does not enlarge the support of $\varphi$.

For any self-adjoint $A$, real $\tau ,\theta$, put

The associated Schrödinger operations are

The operations in (eq:Kraus) form a normal binary CP instrument, and

For normalized $\Psi \in \mathrm{Dom}A$,

For bounded self-adjoint $A,B$,

and the change in a single probability at fixed $\theta$ is at most half this norm.

Proof. Expansion gives ${\sum }_{\pm }{K}_{\pm }^{\ast }{K}_{\pm }=I$, proving normalization. Bounded Kraus maps are normal and CP. The same expansion proves (eq:prob). Integrating the scalar inequality $|e^{ix}-1-ix|\le x^2/2$ against the spectral measure gives (eq:Taylor). The operator bound follows by differentiating $e^{i(\tau -s)A}{e}^{isB}$ and integrating. Formula (eq:prob) gives the factor $1/2$.

The realization is a qubit initially in $|+⟩$, a controlled $U$, and an equatorial qubit readout. This proves the claimed global dilation. It does not prove spacetime localization of the control.

Figure fig:Ramsey-instrument gives the controlled-unitary realization of the common binary instrument.

Let $\mathcal{F}={\Gamma }_s(\mathfrak{h})$, let $\mathsf{N}$ be its number operator, and let ${\mathcal{D}}_{\mathrm{fin}}$ be the union of its finite number sectors. This includes arbitrary vectors in each sector.

Suppose a symmetric operator $A$ preserves ${\mathcal{D}}_{\mathrm{fin}}$, changes particle number by at most two, and satisfies

Then $A$ is essentially self-adjoint on ${\mathcal{D}}_{\mathrm{fin}}$. If also $\Vert A\Psi \Vert \le C\Vert (\mathsf{N}+2)\Psi \Vert$, its closure is defined on $\mathrm{Dom}\mathsf{N}$ and obeys that estimate there.

Proof. Iteration gives

Thus the power series $U(t)\Psi ={\sum }_j(it{)}^j{A}^j\Psi /j!$ converges for $2C|t|<1$, with a radius independent of $n$.

We give the analytic-vector argument explicitly. In a smaller interval, the differentiated double series for $⟨U(t)\Psi ,U(t)\mathrm{X}⟩$ is absolutely convergent. Symmetry cancels its derivative term by term, so $U(t)$ is an isometry on the dense finite-particle space and extends to an isometry on $\mathcal{F}$. For $2C(|s|+|t|)<1$, the double series for $U(s)U(t)\Psi$ converges absolutely by the preceding bound and the binomial identity. It equals $U(s+t)\Psi$. Hence $U(t)$ is unitary for sufficiently small $t$, with inverse $U(-t)$.

If $A^{\ast }h=ih$, then symmetry and the series give

Density implies $U(t{)}^{\ast }h=e^{t}h$, impossible for a nonzero $h$ and a nonzero small real $t$. The same argument excludes $A^{\ast }h=-ih$. Therefore both deficiency spaces vanish. Indeed, the ranges of the closure of $A\pm i$ are closed by ${\Vert (A\pm i)\Psi \Vert }^2={\Vert A\Psi \Vert }^2+{\Vert \Psi \Vert }^2$; their orthogonal complements are those deficiency spaces. Surjectivity of both ranges proves self-adjointness of the closure.

Finally, number truncations converge in the number graph norm. The assumed graph bound makes their $A$-images Cauchy. Closedness proves the last assertion.

This lemma will be used to specify the uncompressed generator of Model 1. For Model 2 the measured dynamics will instead be defined only by its bounded compressions.

Figure fig:finite-band-domain summarizes the finite-band analytic-vector argument used for the Model 1 generator.

For $p\ne 0$, define

It is the orthogonal projector onto symmetric transverse traceless spatial tensors.

Let $p_{\ast }=(0,0,1)$, $H=\mathrm{diag}(1,-1,0)$, and choose a nonnegative even smooth bump $\chi$ supported sufficiently near $p_{\ast }$ and $-p_{\ast }$, away from zero. Put

Choose $\chi$ nonzero where $\Pi (p)H\ne 0$. Then $A$ is real Schwartz and TT, and

In particular $A(0)\ne 0$.

Let $D_0=\sqrt{-\Delta }$, $M_A=\int A_{ij}{A}_{ij}\mathrm{d}x$, and set

These waves are Schwartz on finite time slabs.

With

one has ${\sigma }_{\mathrm{plane}}(e,r)=1$. Every $k=qe+\pi r$ is a linearized vacuum solution, and no nonzero such $k$ is pure gauge.

For

the decomposition $h=k+K\xi$ is unique, and the retarded harmonic map

satisfies $w(h)=\xi$ and $h-Kw(h)=k$.

Proof. The Cauchy data are $(e,\dot{e})=(A,0)$ and $(r,\dot{r})=(0,A/M_A)$, giving the symplectic pairing. TT wave evolution gives $D(k)=0$, zero trace, and $\square k=0$; substitution into the linearized Einstein tensor gives vacuum.

Linearized curvature is unchanged by $K\xi$. For a temporal-gauge TT field, ${R^{(1)}}_{0i0j}(k)=-\frac{1}{2}{{\partial }_t}^2{k}_{ij}$. A pure gauge member of the plane would therefore satisfy $\Delta k=0$. A harmonic Schwartz function vanishes, so $k=0$.

For the section, (eq:gauge-identities) gives $D(h)=\square {\xi }^{♭}$. Since $\xi$ is compactly supported, the retarded zero-past solution of that equation is ${\xi }^{♭}$. A retarded solution can be constructed using zero Cauchy data before the compact source; Lemma lem:wave with $m=0$ gives existence and uniqueness. Uniqueness of $k$ follows from the curvature argument, and a compactly supported Minkowski Killing field is zero. Thus the decomposition and the section are unique.

The displayed symplectic normalization belongs to this plane model. It will not be silently substituted for the distinct normalization of the dense-TT model.

Figure fig:fixed-TT-plane displays the canonical plane and the unique section on its admitted compact-gauge extension.

Let ${\varphi }_0$ be a free scalar with smooth compact initial data. Lemma lem:wave defines $S_{t_0}(h){\varphi }_0$, with support in the causal evolution of those data on each side of $t_0$.

If $H_{t_0}(u)$ denotes the homogeneous scalar solution with the Cauchy data of $u$ at $t_0$, then

Both sides have zero data and solve $P\psi =P(X_{\xi }{\varphi }_0)$, by (eq:gauge-identities); uniqueness proves the equality.

If $S_{t_0}(k){\varphi }_0=0$ for every compact-data free scalar and $k\in \mathrm{span}\{e,r\}$, then $k=0$.

Proof. The field equation gives $k^{ij}{\partial }_i{\partial }_j{\varphi }_0=0$ for every such solution. At any chosen time and point, arbitrary spatial Hessians can be prescribed using compact smooth position data there. Propagating those data to $t_0$ preserves compactness. Therefore $k$ pairs to zero with every symmetric matrix at every point.

For completeness, the admitted first-order family consists of

The complete coefficient expression

is unchanged by adding a compact gauge parameter, since both $w(h)$ and ${\varphi }_1$ change by that parameter's prescribed action. On the selected section it is

No sourced second metric coefficient occurs here.

Take real $f\in {C_c}^{\infty }((t_0,T)\times {\mathbb{R}}^3)$, and define

This sign convention is fixed throughout the paper: $({{\partial }_s}^2+{{\Omega }_m}^2)v_f=f$, with zero terminal data. It differs by a minus sign from the advanced inverse of $P$.

Exchanging the finite-time integrals gives, for TT $a$,

Consequently set

Then ${\ell }_{a,f}=\int g_{a,f}{\varphi }_0$. The spatial support on the finite slab is compact, all spatial derivatives are bounded, and the only possible spacetime nonsmoothness is the initial-time edge.

For such a real $g$, define

Spatial smoothness and finite propagation imply ${\alpha }_g,{\beta }_g\in {C_c}^{\infty }({\mathbb{R}}^3)$. If $(\phi ,\varpi )$ are the scalar data,

Define

Since $m>0$, these vectors are Schwartz. Write $u_{a,f}=u_{g_{a,f}}$.

For real $g,h$,

Thus ${\ell }_g\mapsto \Phi (u_g)$ has the canonical commutator for scalar initial linear functionals.

Proof. Expand (eq:ug). The two diagonal terms have real inner products for real data. Plancherel's identity identifies the imaginary cross terms with half the displayed expression.

Declare the plane Poisson bracket by $\{q,\pi \}=1$, the scalar bracket by Lemma lem:scalarCCR, and mixed brackets zero. This is a specified initial-data bracket, not the reduced bracket of a coupled Einstein–scalar theory. The coefficient quantization

satisfies

Indeed, the surviving terms contain only one scalar CCR; brackets between two correction terms carry ${\kappa }^2$.

On

define initially on the total finite-particle domain

The oscillator is identified with a single Fock mode: $Q=(b+b^{†})/\sqrt{2}$, ${\Pi }_0=-i(b-b^{†})/\sqrt{2}$.

For every real $\lambda$, (eq:A1) is essentially self-adjoint on the total finite-particle domain. Its closure is defined on $\mathrm{Dom}\mathsf{N}$. It acts only on the gravitational oscillator and

in the scalar one-particle space, and is the identity on the orthogonal Fock factor.

Proof. The standard sector estimates are

Products of two such operators obey a constant multiple of $\Vert (\mathsf{N}+2)\Psi \Vert$. Expanding the two mixed terms in (eq:A1) gives a symmetric operator, changing number by at most two, with the estimates of Lemma lem:analytic. Its factors commute because the species differ. The lemma proves essential self-adjointness and the domain inclusion.

Fock factorization over $V_f\oplus {V_f}^{\perp }$ gives the active factor assertion first on finite-particle vectors and then for the unique self-adjoint closure.

There is no evaluation homomorphism sending the nilpotent $\kappa$ in (eq:Fplane) to a nonzero real $\lambda$. Equation (eq:A1) is a separately defined numerical polynomial using the same coefficients. Its spectral functions are predictions of this named model, not higher-order predictions of a formal gravitational expansion.

Figure fig:scalar-to-generator traces the coefficient construction while separating formal correspondence from numerical spectral calculus.

Choose $0\le \vartheta \le 1$, smooth, with $\vartheta (s)=0$ for $s\le 0$ and $1$ for $s\ge 1$. Set

and leave $g_{0,f}$ unchanged. For $0<\epsilon \le 1$, these are genuine compact smooth spacetime smearings. Put

Equation (eq:ug) and Minkowski's integral inequality give

Let ${A^{(1)}}_{f,\epsilon }(\lambda )$ use these regulated vectors. All these operators satisfy the preceding proof with a common constant

in the number estimates. Their difference has the same estimates with

For every fixed real $\tau ,\lambda$,

If $2\overline{C}|\tau |<1$, then on particle number at most $n$,

For a total coherent vector of mean number $\mu$, an upper bound is

which tends to zero.

Proof. On finite-particle vectors use the convergent exponential series from Lemma lem:analytic. Telescoping the $j$-th powers gives the bound

Summing with coefficients $|\tau {|}^j/j!$, and using

proves (eq:edge-unitary). Density and the norm bound two on a difference of unitaries give strong convergence for small times. For arbitrary $\tau$, divide it into finitely many sufficiently small steps and use the group law and strong convergence of uniformly bounded products.

For a coherent vector, truncate at number $n$, use (eq:edge-unitary) on the truncation, and bound the two tail images by twice the tail norm. This gives (eq:Ecoh). First choose $n$ and then let $\epsilon \to 0$.

Figure fig:edge-removal shows the coefficient estimate and the truncation argument behind strong unitary edge removal.

Let $A_0=A(0)$, $c_0=|A_0{|}^2>0$. Choose a real compact bump ${\eta }_0=1$ near zero and take scalar data

Let ${\varphi }_{\ast }$ be the free solution and ${\psi }_{\ast }=S_{t_0}(e){\varphi }_{\ast }$. Its zero correction data imply

Define finite constants on $0\le s\le 1,|x|\le 1$:

Set

Then

for $0\le s\le \delta ,|x|\le r_0$.

Choose nonnegative unit-integral bumps ${\beta }_t\in {C_c}^{\infty }((1/2,1))$ and ${\beta }_x\in {C_c}^{\infty }(B(0,1))$, and define